Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds

Abstract

We show that if the curvature of a Cartan-Hadamard $n$-manifold is constant near a convex hypersurface $\Gamma$, then the total Gauss-Kronecker curvature $\mathcal{G}(\Gamma )$ is not less than that of any convex hypersurface nested inside $\Gamma$. This extends Borbély's monotonicity theorem in hyperbolic space. It follows that $\mathcal{G}(\Gamma )$ is bounded below by the volume of the unit sphere in Euclidean space ${\mathbf{R}}^n$.

7 pages